2. A particle of mass m moves in a straight line under the action of a conservative force F(x)with potential energyU(x) = x²e¯x.(i) Calculate F(x) and find the two equilibrium points of the system. Compute if the equi-libria are stable or unstable. Sketch the potential energy as a function of x, indicating theequilibria on your plot.(ii) Calculate the total mechanical energy E of the system, in terms of v and x. Show thatdE/dt = 0, i.e., the total energy is constant during motion.(hint: use the equation of motion mi=(iii) Assume the particle starts in xoenergy Eo of the particle. Using (ii),with=F)O with positive initial velocity vo > 0. Find the initialshow that the particle reaches x = 2 only if vo > û,8e-2="mand in this case the particle's velocity in x =2 is8e-2v(2):= vm(iv) Assume the particle starts in x0 = 0 with positive initial velocity vo > û. Use (ii) tofind the expression for v(x) and find the terminal velocity of the particle as x → ∞. If theparticle starts with negative initial velocity vo < 0, can it escape to x →∞o?(v) Assume m=1, show that the equation of motion isd²xd+2=-= x(x − 2)e¯x.

The objective of the question is to analyze the motion of a particle under a conservative force with…

Answered: 2. A particle of mass m moves in a…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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H2. A particle of mass m moves in a straight line under the action of a conservative force F(x) with potential energy U(x) = x²e-*. (i) Calculate F(x) and find the two equilibrium points of the system (i.e., points xẻ such that F(x) = 0). Compute if the equilibria are stable (i.e., local minima of the potential energy: U" (xe) > 0) or unstable (i.e., local maxima of the potential energy: U"(xe) 0. Find the initial energy Eo of the particle. Using (ii), show that the particle reaches x = 2 only if vo > ŷ, with 8e-2 v= and in this case the particle's velocity in x = 2 is v(2): = (a) x = 0, vo=0 (b) x=2, vo = 0 (c) x = 0, vo= 0.5 (d) x = 0, vo = -0.5 (e) x = 0, vo=2 (f) x = 0, vo= -10. m 8e-2 v². m (iv) Assume the particle starts in xo =0 with positive initial velocity vo > ŷ. Use (ii) to find the expression for v(x) and find the terminal velocity of the particle as x → ∞. If the particle starts with negative initial velocity vo < 0, can it escape to x →→∞? (v) Assume m= 1, show that the…
what are the stable equilibria (X1 ,X2)?
(3) Let h(x,y) = r'e. Find its linearization at the point (1,0) and use this to approximate f(2,0). %3D
2 and 3
2. A particle of mass m moves in a straight line under the action of a conservative force F(x) with potential energy U(x) = = x²ex. (i) Calculate F(x) and find the two equilibrium points of the system. Compute if the equi- libria are stable or unstable. Sketch the potential energy as a function of x, indicating the equilibria on your plot. (ii) Calculate the total mechanical energy E of the system, in terms of vand x. Show that dE/dt = 0, i.e., the total energy is constant during motion. (hint: use the equation of motion mv = F) (iii) Assume the particle starts in x0 = 0 with positive initial velocity vo > 0. Find the initial energy Eo of the particle. Using (ii), show that the particle reaches x = 2 only if vo > ô, with 8e-2 v = m and in this case the particle's velocity in x = 2 is 8e-2 ༧(2) vo m (iv) Assume the particle starts in x0 = 0 with positive initial velocity vo > v. Use (ii) to find the expression for v(x) and find the terminal velocity of the particle as x → ∞. If the…
Consider a theoretical model for mutual inhibition in ecological systems where n > 0 measures the strength of mutual inhibition d dt x(t) = (1/2)" (1/2)" +yn - x d dt (t) = (1/2)" - (1/2)" +x" -y Plot the nullclines and determine the equilibria points for n = 1 and n = 3. Interpret these equilibrium in the ecological setting. Explicitly calculate the local stability of the equilibrium point where x* = y* for gen- eral n. When does the system undergo a qualitative change in behaviour?
3. A projectile is fired with an initial velocity of 128 ft/s from a cannon at time t = 0 from a height of 32 ft above the ground and at an angle of 30° with the horizontal. (a) Determine: x '(0) and y'(0). (These values represent the horizontal and vertical components of the initial velocity vector.) (b) Use integration to determine the position equations: x(t) and y(t) for the projectile t seconds after being fired by assuming that the projectile is in freefall. Thus, x"(t)=0 ft/s² and y"(t)=-32 ft/s² for all t≥ 0. (Note: x"(t) and y"(t) are the horizontal & vertical components of the projectile's acceleration vector at time t.) (c) Determine the maximum height, H, of the projectile and its horizontal distance, D, from the launch site at the time that it achieves this height. (Hint: The maximal height is attained at the time, t*, at which y'(t*) = 0.) (d) Determine the range of the projectile, R. YA (0,32) - P. 32' 128 0=30° x'(0) y'(0) H R Pr=(x(t), y(t)) is the position of the…
5. Okay, time for some mathematics. We're going to focus on the interactions between the hares and their main predators, the lynx. The tool we need is differential equations (last seen when we talked about the S-I-R disease model). We will let H(t) be the density of hares at time t, and L(t) the density of lynx at time t. Our first assumption is that, if there are no lynx around, the hares grow exponentially. Hares lead to more hares and faster growth. This gives us the differential equation (trust me): HP = aH dt where a is a positive constant. Second we assume that, if there are no hares around, the lynx starve and thus their numbers decay exponentially. This gives us: TP = -bL dt
3. A big slug weighing 10 pounds stretches a spring ½ foot. The slug is removed and replaced by another slug, of mass 1.6 slugs. The slug on the spring is then taken to the bottom of the sea (where the water offers resistance that is approximately equal to the velocity of the object) and then released from a position of 1/3 foot above the equilibrium with an downward velocity of 5/4 feet per second. (a) Set up the DE for this situation and show all the steps to find the position equation for the spring. Use X for the position and t for the time. Round all values to 2 decimal places for simplicity. (b) Graph your equation from (a), labeling the axes, and use your graph to find the second time where the slug is heading downwards and passes equillibrium. Write a sentence to summarize.
Given the equation of state of a PVT system is bT² √P Where b is a constant. V = (a) Calculate the thermal expansivity B =) - = (3/P) ₁ 1 (c) Express the total differential of V in terms of ß and K. (b) Calculate the compressibility K =

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